A man travels $ m$ feet due north at $ 2$ minutes per mile. He returns due south to his starting point at $ 2$ miles per minute. The average rate in miles per hour for the entire trip is: $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 24\qquad\\ \textbf{(E)}\ \text{impossible to determine without knowing the value of }{m}$

In my calculator, one of the keys from $1$ to $9$ does not work properly: when you press it, a digit between $1$ and $9$ appears on the screen that is not the correct one. When I tried to write the number $987654321$, a number divisible by $11$ appeared on the screen and leaves a remainder of $3$ when divided by $9$. What is the broken key? What is the number that appeared on the screen?

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$

[b]3/1/32.[/b] The bisectors of the internal angles of parallelogram $ABCD$ determine a quadrilateral with the same area as $ABCD$. Given that $AB > BC$, compute, with proof, the ratio $\frac{AB}{BC}$.

Given an infinite sequence of numbers $a_1, a_2,...$, in which there are no two equal members. Segment $a_i, a_{i+1}, ..., a_{i+m-1}$ of this sequence is called a monotone segment of length $m$, if $a_i < a_{i+1} <...<a_{i+m-1}$ or $a_i > a_{i+1} >... > a_{i+m-1}$. It turned out that for each natural $k$ the term $a_k$ is contained in some monotonic segment of length $k + 1$. Prove that there exists a natural $N$ such that the sequence $a_N , a_{N+1} ,...$ monotonic.

Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$. The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$. If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw(M--N^^C--A--P--B--C^^Arc(origin,1,0,180)); markscalefactor=0.03; draw(anglemark(C,A,P)); draw(anglemark(C,B,P)); pair point=C; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, S); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$P$", P, S); label("$40^\circ$", C+(-0.15,0), NW); label("$10^\circ$", B+(0,0.05), W); label("$10^\circ$", A+(0.05,0.02), E);[/asy] $ \textbf{(A)}\ 10^{\circ}\qquad\textbf{(B)}\ 15^{\circ}\qquad\textbf{(C)}\ 20^{\circ}\qquad\textbf{(D)}\ 25^{\circ}\qquad\textbf{(E)}\ 30^{\circ}$

The points $M$ and $N$ are on the side $BC$ of $\Delta ABC$, so that $BM=CN$ and $M$ is between $B$ and $N$. Points $P\in AN$ and $Q\in AM$ are such that $\angle PMC=\angle MAB$ and $\angle QNB=\angle NAC$. Prove that $\angle QBC=\angle PCB$.

Consider an infinite sequence of reals $x_1, x_2, x_3, ...$ such that $x_1 = 1$, $x_2 =\frac{2\sqrt3}{3}$ and with the recursive relationship $$n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.$$ Find $x_{2019}$.

There are $26$ students in the class. They agreed that each of them would either be a liar (liars always lie) or a knight (knights always tell the truth). When they came to the class and sat down for desks, each of them said: “I am sitting next to a liar.” Then some students moved for other desks. After that, everyone says: “ I am sitting next to a knight .” Is this possible? Every time exactly two students sat at any desk.

Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$ then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle. by Yu.Blinkov

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

The incircle of $ \triangle A_{0}B_{0}C_{0}$, meets legs $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$, respectively on points $A$, $B$, $C$, and the incircle of $ \triangle ABC$, with center $I$, meets legs $BC$, $CA$, $AB$, on points $A_{1}$, $B_{1}$, $C_{1}$, respectively. We write with $ \sigma (ABC)$, and $ \sigma (A_{1}B_{1}C_{1})$ the areas of $ \triangle ABC$, and $ \triangle A_{1}B_{1}C_{1}$ respectively. Prove that if $ \sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})$, then lines $AA_{0}$, $BB_{0}$, $CC_{0}$ are concurrent.

How many zeros does $101^{100} - 1$ end with?

Determine all natural numbers $m$ and $n$ such that \[ n \cdot (n + 1) = 3^m + s(n) + 1182, \] where $s(n)$ represents the sum of the digits of the natural number $n$.

A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\] At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$; $c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists. At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$. Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.

In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b. [asy] pair A = (0,3); pair B = (0,0); pair C = (3,0); pair D = (0,1.5); pair E = (0.35,0); pair F = (1.2,1.8); pair J = (0.17,0); pair Y = (0.17,0.75); pair Z = (1.6,0.2); draw(A--B); draw(B--C); draw(C--A); draw(D--F--Z--E--D); draw("$O$", B, dir(180)); draw("$B$", A, dir(45)); draw("$A$", C, dir(45)); draw("$Q$", E, dir(45)); draw("$P$", D, dir(45)); draw("$R$", Z, dir(45)); draw("$S$", F, dir(45)); draw("$a$", Y, dir(210)); draw("$b$", J, dir(100)); [/asy]

Given positive integers $m,n(2\le m\le n)$, let $a_1,a_2,\ldots ,a_m$ be a permutation of any $m$ pairwise distinct numbers taken from $1,2,\ldots ,n$. If there exist $k\in\{1,2,\ldots ,m\}$ such that $a_k+k$ is odd, or there exist positive integers $k,l(1\le k<l\le m)$ such that $a_k>a_l$, then call $a_1,a_2,\ldots ,a_m$ a [i]good[/i] sequence. Find the number of good sequences.

Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$.

Convex quadrilateral $ABCD$ is inscribed in circle $w.$Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$ [i](Kungozhin M.)[/i]